WASHINGTON >> A grand jury’s indictment Monday of two abortion opponents who covertly recorded Planned Parenthood officials is the latest, most startling sign that a Republican campaign against the group has run into trouble.

In a dozen states including Texas, where the Houston grand jury examined Planned Parenthood at the request of Republican officials but ended up indicting the opponents, various investigations have concluded without finding any wrongdoing by affiliates of the group. Eight states have declined to investigate since videos began surfacing in June alleging that Planned Parenthood illegally sells tissue from aborted fetuses.

In the Republican-led Congress, Speaker John A. Boehner resigned last fall rather than lead a government shutdown to force an end to federal funds for Planned Parenthood. Conservatives’ efforts to defund the group have since failed. Senate leaders increasingly fear the fight threatens not only several Republican seats but also the party’s majority. Five congressional committees investigating the organization have yet to produce results.

In the presidential race, Republican candidates are competing to condemn Planned Parenthood in an effort to appeal to conservative voters. Sen. Ted Cruz of Texas, a favorite of many evangelical conservatives, has regularly vowed that as president he will order a federal investigation into the group’s practices and has lambasted congressional Republican leaders for timidity. Party strategists worry such attacks will backfire with the general electorate in November, especially among women, younger voters and political independents.

Together these developments “highlight the political problem that could be caused if we were to nominate someone like Ted Cruz at the top of the ticket,” said Brian Walsh, a Republican consultant and former adviser to House and Senate leaders, who has not endorsed a presidential candidate.

“In the last several elections, the Democratic playbook has been to discuss the ‘war on women’ narrative” against Republicans, Walsh said. He added, “If you have a much more polarizing ideologue like Ted Cruz at the top of the ticket, you would make it that much easier for Democrats to prosecute that argument.”

Karl Rove, a Republican strategist and a former top adviser to President George W. Bush, has called proponents of shutting down the government to defund Planned Parenthood the “suicide caucus”; party elders are well aware that Republican attacks on the organization back to the Clinton era have not ended well for Republicans. On Tuesday, the House Democrats’ campaign organization attacked Republicans in 18 House races, asking in news releases whether the candidates still supported the “now indicted conspirators behind Planned Parenthood videos.”

Nonpartisan opinion polls suggest Republicans are right to be concerned.

A majority of Americans continue to support Planned Parenthood and its federal payments, which reimburse nearly 700 affiliates for providing reproductive care, preventive health services and treatment of sexually transmitted diseases to low-income Medicaid recipients. Payments for abortions, which are performed by just over half of Planned Parenthood centers, are prohibited by federal law except in cases of rape, incest and a pregnancy’s threat to a woman’s life.

A survey for The New York Times and CBS News this month showed that nearly 6 in 10 Americans say Planned Parenthood should receive federal funds. That finding was statistically unchanged from a similar survey in September, even as conservatives at the local, state and federal levels stoked outrage about the videos from a group called Center for Medical Progress founded by a 27-year-old Californian, David R. Daleiden, one of those indicted on Monday.

Daleiden and Sandra S. Merritt, 62, were indicted on felony charges of tampering with a governmental record with the intent to defraud — specifically, falsifying California driver’s licenses to pose as biotechnology representatives and infiltrate Planned Parenthood centers and research conferences. Daleiden also was charged with a misdemeanor related to trying to buy human organs.

The Times/CBS poll and others show that more people continue to hold favorable opinions than unfavorable ones about the organization. One in 5 American women visits a Planned Parenthood center at least once in her lifetime, according to the organization, while the Guttmacher Institute, which researches reproduction issues, says 3 in 10 American women have had an abortion by the age of 45.

The attack against Planned Parenthood “has been an abject failure in terms of public opinion,” said Geoff Garin, a pollster for Democrats and for Planned Parenthood.

“Having an extreme anti-abortion group as the source of the videos has been an important factor in undermining their credibility and impact,” he added. “And the Harris County indictments undoubtedly will reinforce that.”

Planned Parenthood’s legal counsel first suggested to Republican congressional investigators in July — after Daleiden posted the first in a monthslong series of videos online — that he might be culpable not only of identity fraud but also of violations of state and federal laws related to corporate filings and tax-exempt organizations.

Congressional Democrats, led by Rep. Jan Schakowsky of Illinois, similarly have alleged that Daleiden and his associates probably broke laws and called on the Justice Department and the California attorney general to investigate.

Daleiden has said from the start, and repeated Monday, that he acted legally, as a “citizen journalist.”

He also has acknowledged in interviews that since 2013 he incorporated a phony company in California called Biomax Procurement Services, created the tax-exempt Center for Medical Progress and its website, and then secretly filmed his encounters as he gained access to Planned Parenthood centers in several states, including California, Colorado and Texas.

According to Planned Parenthood, few affiliates have had arrangements to provide tissue from aborted fetuses to researchers or companies that acquire tissue for researchers, partly because of the potential for controversy. Only affiliates in California, Oregon and Washington have fetal tissue transfer programs.

Planned Parenthood said that independent affiliates do not profit, which would be illegal, but that they charge only for handling costs. In the video, an official suggested a range of $30 to $100 a specimen; the phony representatives offered up to $1,600.

A special committee created by House Republicans to investigate abortion and fetal tissue practices is just getting started this month. Its chairwoman, Rep. Marsha Blackburn, R-Tenn., said it will not report its findings until December — after the November elections. In a statement Tuesday, she said the grand jury indictments would have no bearing on her committee’s work.

“The lack of respect for infant lives on display in the videos is just as abhorrent today as it was yesterday,” she said, adding, “We will continue to gather information and get the facts about medical practices of abortion service providers and the business practices of the procurement organizations who sell baby body parts.”